ESAIM: Control, Optimisation and Calculus of Variations July 2005, Vol. 11, 382–400
DOI: 10.1051/cocv:2005011
CONICAL DIFFERENTIABILITY FOR BONE REMODELING CONTACT ROD
MODELS
∗Isabel N. Figueiredo
1, Carlos F. Leal
1and Cec´
ılia S. Pinto
2Abstract. We prove the conical differentiability of the solution to a bone remodeling contact rod model, for given data (applied loads and rigid obstacle), with respect to small perturbations of the cross section of the rod. The proof is based on the special structure of the model, composed of a variational inequality coupled with an ordinary differential equation with respect to time. This structure enables the verification of the two following fundamental results: the polyhedricity of a modified displacement constraint set defined by the obstacle and the differentiability of the two forms associated to the variational inequality.
Mathematics Subject Classification. 49J15, 49J20, 74B20, 74K10, 74L15, 90C31. Received June 29, 2004.
Introduction
We consider a bone remodeling model, for a rod that may come into contact without friction with a rigid obstacle, due to the action of external loads, and we characterize the conical differentiability of the solution to this model, with respect to small variations of the geometry of the cross section of the rod. The knowledge of this conical differentiability is important, for example, in shape optimization bone remodeling problems, where the purpose is to control the geometry of the rod. In this introduction we describe the model and summarize the essential results of this paper.
Let s ∈ [0, δ], δ > 0, be a small parameter and Ωs = ωs×]0, L[ a domain, representing the reference
configuration of a rod with cross section ωs⊂ R2and axis length L > 0. For each s∈ [0, δ], ωs= ω + sθ(ω) is
a perturbation of ω ⊂ R2 in the direction of the vector field θ = (θ
1, θ2) :R2 −→ R2, that is regular enough. Consequently, the set Ωsis a perturbation of the rod Ω = Ω0= ω×]0, L[. Let V be a Hilbert space, representing the admissible displacements of the rod and K ⊂ V a convex and closed subset of V , defining the constraints imposed on the admissible displacements of the rod. This set K represents the possible contact, without friction, of the rod with the rigid obstacle. Let < ., > denote the duality between V and V , where V is the dual of V , let x = (x1, x2, x3) be a generic element of Ω, and let t be the time variable in the interval [0, T ], with T > 0 a
Keywords and phrases. Adaptive elasticity, functional spaces, polyhedric set, rod.
∗This work is part of the project “New materials, adaptive systems and their nonlinearities; modelling, control and numerical
simulation” carried out in the framework of the european community program “Improving the human research potential and the socio-economic knowledge base” (HRN-CT-2002-00284).
1 Departamento de Matem´atica, Universidade de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal;
isabel.figueiredo@mat.uc.pt; carlosl@mat.uc.pt
2 Departamento de Matem´atica, Escola Superior de Tecnologia de Viseu, Campus Polit´ecnico 3504-510 Viseu, Portugal;
cagostinho@mat.estv.ipv.pt
c
EDP Sciences, SMAI 2005
real positive parameter. Given a function gs(x, t), depending on s∈ [0, δ] and defined in Ω × [0, T ], we denote
by ˙gsand ∂
igsits partial derivatives, with respect to time t and to xi for i = 1, 2, 3, respectively.
For each perturbed rod Ωs, with s ∈ [0, δ], the bone remodeling rod model that we consider is the
elas-tic adaptive reduced rod model derived by Figueiredo and Trabucho [5], but with different boundary condi-tions and additional constraints (we recall that the theory of adaptive elasticity was introduced by Cowin and Hegedus [2, 7] and describes the physiological process of bone remodeling, that is, the continual process of growth, reinforcement, deposition and absorption of material, which occurs in living bone). Moreover, the bone remodeling model that we adopt in this paper, can be mathematically justified by the asymptotic expansion method as in Figueiredo and Trabucho [5] (cf. also Trabucho and Vi˜ano [13], for an explanation of the as-ymptotic expansion method applied to elastic rod contact models), and is defined by the following system, formulated in the set Ω× [0, T ] independent of s (cf. (1.9))
Find (us, ds) such that:
us= (us1, us2, us3) : Ω× [0, T ] → R3 and ds: Ω× [0, T ] → R, us(., t)∈ K ⊂ V, ads(us, v− us) ≥ Lds, v− us, ∀v ∈ K ⊂ V, ˙ ds= h(s, θ, ds, us), in Ω× (0, T ), ds(x, 0) = ds(x), in Ω. (0.1)
The pair (us, ds) is the unknown of the model: the vector field us(., t) represents the displacement of the rod Ω
at time t and the scalar field ds(., t) is the measure of change in volume fraction of the elastic material of the
rod Ω at time t (from a reference volume fraction of elastic material present in the porous bone, identified with the set Ω). The variational inequality, where ads(., .) is a bilinear form and Ldsa linear form that depend on ds,
expresses the equilibrium of the rod Ω under the action of external forces, and subjected to the displacement constraints defined by the set K, that represents the possible contact of the rod with the rigid obstacle. The ordinary differential equation with respect to time, where h is a function that depends on us, ds, θ and s
(cf. (1.9) and (1.17)), is the so-called remodeling rate equation and models the physiological process of bone remodeling – if ˙ds is positive (respectively negative) it means that the volume fraction of elastic material is
increasing (respectively decreasing). The unknowns us and ds are interdependent: the displacement us is the solution of the variational inequality and depends on dsand the unknown ds depends on usand is the solution
of the ordinary differential equation with respect to time.
The aim is to analyze the right-derivative of the solution to problem (0.1), with respect to s, at s = 0. To compute this derivative we mainly use the regularity hypotheses for the solution to problem (0.1), convenient a priori norm bound estimates for the families {(us, ds)}s>0 and {(u
s−u0
s , ds−d0
s )}s>0, where (u0, d0) is the
solution to problem (0.1) with s = 0, Theorem 4.14 of Sokolowski and Zolesio [12], p. 178 (or equivalentely, Th. 4.30 of Sokolowski and Zolesio [12] p. 210), the Schauder’s fixed point theorem and uniqueness results. We remark that, in order to be able to apply the above mentioned Theorem 4.14 of Sokolowski and Zolesio [12], p. 178, we prove the polyhedricity of a modified constraint displacement set, using a technique described in Sokolowski and Zolesio [12], p. 209, and assuming an appropriate additional condition imposed to a non-negative Radon measure, as indicated in Proposition 3.4.
The main theorem of the paper can be formulated as follows.
Theorem 0.1. For each t ∈ [0, T ], let As(., t)∈ L(V ; V) be the linear operator defined by Asv, u = ads(v, u)
for all v, u in V . Then the following three statements i), ii) and iii) are verified. i) For each t∈ [0, T ], there exists A(., t)∈ L(V ; V) such that
lim s→0+ As− A0 s − A (., t) L(V ;V) = 0. (0.2)
ii) For each t∈ [0, T ], there exists L(., t)∈ V such that lim s→0+ Lds− Ld0 s − L (., t) V = 0. (0.3)
iii) With the hypothesis of Proposition 3.4, for each time t∈ [0, T ] the solution Π(Ld0)(., t) of the variational
inequality
u0(., t) = Π(L
d0)(., t),
ad0(u0, v− u0) ≥ Ld0, v− u0, ∀v ∈ K ⊂ V,
(0.4) is conical differentiable at Ld0(., t), that is
∀l ∈ V, Π(L
d0+ sl)(., t) = Π(Ld0)(., t) + sQ(l)(., t) + o(s) (0.5)
for all s > 0, small enough, where for each t, the mapping Q(., t) : V → V is continuous and positively homogeneous and o(s)V
s → 0, as s → 0+.
Consequently, the properties i), ii) and iii) imply that, for each t ∈ [0, T ], the solution (us, ds)(., t) to the
problem (0.1) is right-differentiable with respect to s, at s = 0
us(., t) = u0(., t) + su(., t) + o(s), in V, and u= Q(L− Au0), (0.6) ds(., t) = d0(., t) + sd(., t) + r(s), in L2(Ω), (0.7) for all s > 0, small enough, where (u0, d0) is the solution of (0.1) for s = 0 and as s→ 0+, o(s)V
s → 0 and
Ωr(s)vdΩ
s → 0, for any v ∈ L2(Ω).
In particular Aand Lare defined by (2.25) and (2.26), Q is defined by (3.28) and the pair (u, d) is the unique solution of problem (5.1).
Finally let us briefly explain the contents of this paper. In Section 1 we introduce the family of bone remodeling rod models. In Sections 2 and 4 we give partial proofs of the conditions (0.2)–(0.3) and (0.6)–(0.7), respectively. In Section 3 we prove the property (0.5). Finally in Section 5 we completely prove Theorem 0.1.
1. The family of rod models
In this section we introduce some notations, definitions and hypotheses, we define the family of rod models depending on the parameter s, we redefine this family on a set independent of s and finally we give some results concerning the existence and uniqueness of solution.
1.1. Notations, definitions and hypotheses
Let δ > 0 be a small parameter and for each s∈ [0, δ] we consider the perturbation Isof the identity operator I
inR2, defined by I
s(x1, x2) = (I +sθ)(x1, x2) = (xs1, xs2), for all (x1, x2)∈ R2, where θ = (θ1, θ2) :R2−→ R2is a vector field regular enough (at least θ∈ [W2,∞(R2)]2). Let ω be an open, bounded and connected subset ofR2, with a boundary ∂ω regular enough. For each s∈ [0, δ] we define ωs= Is(ω), which is the perturbation of ω in
the direction of the vector field θ. We also denote by Ωs the set occupied by a cylindrical rod, in its reference
configuration, with length L > 0 and cross section ωs, that is Ωs= ωs× [0, L] = Is(ω)× [0, L] ⊂ R3. Moreover
we denote by xs= (xs1, xs2, x3) a generic element of Ωs and define the sets Γs= ∂ωs×]0, L[, Γs0= ωs× {0},
ΓsL= ωs× {L}, where ∂ωsis the boundary of ωs. These three sets represent, respectively, the lateral boundary
of the rod Ωs and its extremities. We assume that the boundary ∂ωs is divided into two nonempty disjoint
We assume that, for each s ∈ [0, δ] the coordinate system (O, xs1, xs2, x3) is a principal system of inertia associated with the rod Ωs. Consequently, axis Ox3 passes through the centroid of each section ωs× {x3} and
we have ω
sxs1dωs =
ωsxs2dωs=
ωsxs1xs2dωs= 0 (we observe that the choice of the vector field θ, that
realizes the shape variation of the cross section ω, must be admissible with this last condition). The set Cm(Ω
s) stands for the space of real functions m times continuously differentiable in Ωs. The spaces
Hm(Ω
s) = Wm,2(Ωs) and W0,2(Ωs) = L2(Ωs) are the usual Sobolev spaces, where m≥ 0 is a positive integer.
The norms in these Sobolev spaces are denoted by.Wm,2(Ωs).
Throughout the paper, the latin indices i, j, k, l... belong to the set{1, 2, 3}, the greek indices α, β, µ... vary in the set{1, 2} and the summation convention with respect to repeated indices is employed, that is, for example, aibi=
3
i=1aibi.
Let T > 0 be a real parameter and we denote by t the time variable in the interval [0, T ]. If V is a topological vectorial space, the set Cm([0, T ]; V ) is the space of functions g : t∈ [0, T ] → g(t) ∈ V , such that g is m times
continuously differentiable with respect to t. If V is a Banach space we denote .Cm([0,T ];V ) the usual norm
in Cm([0, T ]; V ). Moreover, given a function g
s(xs, t) defined in Ωs× [0, T ] we denote by ˙gsits partial derivative
with respect to time, by ∂sαgs and ∂3gs its partial derivatives with respect to xsα and x3, that is, ˙gs = ∂g∂ts,
∂sαgs=∂x∂gs
sα and ∂3gs= ∂gs ∂x3·
For each s ∈ [0, δ] we consider the following model for the rod Ωs, that can be mathematically justified by
the asymptotic expansion method as in Figueiredo and Trabucho [5].
Find (us, ds) such that:
us= (us1, us2, us3) : Ωs× [0, T ] → R3 and ds: Ωs× [0, T ] → R, us(., t)∈ Ks⊂ Vs, ads(us, vs− us) ≥ Lds, vs− us, ∀vs∈ Ks⊂ Vs, ˙ ds= b(ds) + c(ds)e33(us), in Ωs× (0, T ), ds(x, 0) = ds(x), in Ωs. (1.1)
The unknowns of the model (1.1) are the displacement vector field us(xs, t), corresponding to the displacement
of the point xs of the rod Ωs at time t and the measure of change in volume fraction of the elastic material
(from a reference volume fraction denoted in the sequel by ξs0) ds(xs, t) at (xs, t). In particular e33(us) is an
element of the linear strain tensoreij(us)
=12(∂siusj+ ∂sjusi)
, and it is a function of us.
On the other hand, the data of the model (1.1) are the following: the space Vs of admissible displacements,
the set Ks⊂ Vsof displacement constraints, the bilinear form ads(., .) : Vs× Vs→ R and the element Lds(.)∈
V, that depend on the unknown ds and represent, respectively, the elastic equilibrium equations and the
external forces acting on the rod, the initial value of the change in volume fraction ds(.) = ds(., 0), and the
coefficients b(ds) and c(ds) which are material coefficients depending upon the change in volume fraction ds.
Assuming that the rod is clamped at its extremities Γs0 = ωs× {0} and ΓsL = ωs× {L}, the space Vs of
admissible displacements is defined by Vs= vs∈ W02,2(]0, L[) 2 × W1,2(Ω s) : eαβ(vs) = e3β(vs) = 0 (1.2) which is identified with the set
vs= (vs1, vs2, vs3)∈ W02,2(]0, L[) 2 × W1,2(Ω s) : vsα(xs) = vsα(x3), vs3(xs) = vs3(x3)− xsα∂3vα(x3), vs3∈ W01,2(]0, L[) , (1.3)
that is, Vs ⊂ [W1,2(Ωs)]3 is the space of Bernoulli-Navier displacements. We remark that W01,2(]0, L[) ={ξ ∈
W1,2(]0, L[) : ξ(0) = ξ(L) = 0}, and W2,2
0 (]0, L[) ={ξ ∈ W2,2(]0, L[) : ξ(0) = ξ(L) = 0, ξ(0) = ξ(L) = 0}, where ξ is the first derivative of ξ.
The bilinear form ads(., .) is defined
ads(us, vs) = Ωs 1 b3333(ds) e33(us) e33(vs)dΩs, ∀us, vs∈ Vs, (1.4)
where e33(vs) = ∂3vs3= ∂3vs3− xsα∂33vsα and b3333(ds) is a material coefficient that depends on ds (in fact
it is an element of the matrixbijkl(ds)
which is the inverse of the matrix composed of the three-dimensional elastic coefficients of the rod Ωs, as explained in Figueiredo and Trabucho [5]).
The element Lds is defined by
Lds, vs = Ωs γ(ξs0+ Pη(ds)) fsivsidΩs+ Γsg gsivsidΓs, ∀vs∈ Vs, (1.5)
where γ is the density of the full elastic material, which is supposed to be a constant independent of s, ξs0 is
the reference volume fraction of the elastic material (already mentioned immediately after the definition of the problem (1.1)) that belongs to C1(Ω
s), fs= (fsi) and g = (gsi) are, respectively, the density of body loads and
normal tractions on the lateral boundary Γsg of the rod Ωs, and Pη(.) is a truncation operator. We suppose
that 0 < ξmin
s0 ≤ ξs0(xs)≤ ξmaxs0 < 1, for all xs∈ Ωs, and the truncation operator Pη is of class C1 and satisfies
0 < η2 ≤ (ξs0+ Pη(ds))(xs) ≤ 1 for all xs ∈ Ωs, where η > 0 is a small parameter. We also assume that
fsi∈ C1([0, T ]) and gsi∈ C1([0, T ]; W1−1/p,p(Γsg)), with p > 3. These hypotheses of regularity on the forces
are necessary to obtain existence results.
The set Ks ⊂ Vs is a nonempty, closed and convex subset of Vs, representing the additional constraints
imposed on the admissible displacements. Due to the action of the applied loads we assume that the lateral surface Γsc of the rod may come into contact, without friction, with a rigid obstacle. Moreover, we suppose
that the candidate contact surface Γscis plane and perpendicular to the inertia axis Oxs1of the rod. Therefore,
from these assumptions we deduce that the set Ksof the reduced elastic adaptive rod model (1.1) is of the form
(cf. also Trabucho and Vi˜ano [13], Chap. VI, p. 770 (28.46))
Ks={vs∈ Vs: vs1≥ ψ in ]0, L[} (1.6)
where ψ : [0, L]→ R is a smooth enough scalar function, such that ψ(x3) < 0, for all x3 ∈ [0, L]. Then the set Ks physically imposes that the bending component vs1, of the admissible displacement vs, can touch but
not penetrate the obstacle represented by the function ψ.
Finally, we suppose that the initial value ds(.) = ds(., 0) of the change in volume fraction verifies ds∈ C0(Ωs)
and the material coefficients b(ds), c(ds) and b3333(ds) appearing in the right hand side of the remodeling rate
equation are continuously differentiable with respect to ds. In addition we also assume that there exist strictly
positive constants C1, C2, C3, C4, C5 and C6 independent of s and t such that for any (xs, t)∈ Ωs× [0, T ]
0≤ C1≤ 1
b3333(ds) ≤ C2
, ∀s ∈ [0, δ],
|b(ds)| ≤ C3, |b(ds)| ≤ C4, |c(ds)| ≤ C5, |c(ds)| ≤ C6, ∀s ∈ [0, δ],
(1.7)
where b(.) and c(.) are the derivatives of the scalar functions b(.) and c(.), respectively.
We observe that we could have considered in (1.1) a remodeling rate equation depending nonlinearly on e33(us),
that is (cf. Figueiredo and Trabucho [5]) ˙
ds= b(ds) + c(ds)e33(us) +
1 b3333(ds)
which is an equation that seems to be more suitable to represent the remodeling rate process, from the mechanical view-point, even in the case of small strains (cf. Hegedus and Cowin [7]). In fact, all the results of Theorem 0.1 can also be derived for this type of nonlinear remodeling rate equation; the nonlinear term 1
b3333(ds)e33(us)e33(us)
in (1.8) only originates more complicated calculus.
1.2. The family of rod models formulated in Ω
In order to derive the results stated in Theorem 0.1 we reformulate now, for each s∈ [0, δ], the problem (1.1) in the fixed rod Ω independent of s.
We consider the perturbation map Isdefined in Section 1.1 that maps Ω onto Ωs. For a function vsdefined
in Ωs we associate the corresponding function vs (with upper index s) defined in Ω by vs= vsoIs. Performing
this change of variables and observing that e33(vs) = e33(vs)− sθα∂33vsα, for any vs∈ Vs, the problem (1.1) is
equivalent to the following problem defined in the rod Ω independent of s
Find (us, ds) such that:
us= (us 1, us2, us3) : Ω× [0, T ] → R3 and ds: Ω× [0, T ] → R, us(., t)∈ K ⊂ V, ads(us, v− us)≥ Lds, v− us, ∀v ∈ K, ˙ ds= b(ds ) + c(ds)e33(us)− sc(ds)θα∂33usα, in Ω× (0, T ), ds(x, 0) = d(x), in Ω, (1.9)
where we suppose that d is independent of s∈ [0, δ], and, for all u and v in V
ads(u, v) = as0(u, v) + sas1(u, v) + s2a2s(u, v) + s3as3(u, v) + s4as4(u, v), (1.10)
and L ds, v = F0s(v) + Gs0(v) + s Fs 1(v) + Gs1(v) +s2Fs 2(v) + Gs2(v) + s3Fs 3(v) + Gs3(v) . (1.11)
The bilinear forms as
i(., .), for i = 0, 1, 2, 3, 4, are defined by
as0(u, v) = Ω 1 b3333(ds)e33(u)e33(v)dΩ, as1(u, v) = Ω 1 b3333(ds) − θα e33(u)∂33vα+ e33(v)∂33uα + e33(u)e33(v)div θ dΩ, as2(u, v) = Ω 1 b3333(ds)
e33(u)e33(v) det∇θ + θαθβ∂33uα∂33vβ− (div θ) θα
e33(u)∂33vα+ e33(v)∂33uα dΩ, as3(u, v) = Ω 1 b3333(ds) θαθβ∂33uα∂33vβdiv θ− (det ∇θ) θα e33(u)∂33vα+ e33(v)∂33uα dΩ, as4(u, v) = Ω 1 b3333(ds) θαθβ∂33uα∂33vβ det∇θ dΩ. (1.12)
The forms Fs
j(v) and Gsj(v), for j = 0, 1, 2, 3, are defined by
F0s(v) = Ω γ(ξs0+ Pη(ds)) (fαsvα+ f3sv3) dΩ, F1s(v) = Ω γ(ξs0+ Pη(ds)) (fαsvα+ f3sv3) div θ− f3sθα∂3vα dΩ, F2s(v) = Ω γ(ξs0+ Pη(ds)) (fαsvα+ f3sv3) det∇θ − f3sθα∂3vαdiv θ dΩ, F3s(v) =− Ω γ(ξ0s+ Pη(ds)) f3sθα∂3vαdet∇θ dΩ, (1.13) and Gs0(v) = Γg gαsvα+ g3sv3 dΓ, Gs1(v) = Γg (gαsvα+ g3sv3) G1(θ, n)− g3sθα∂3vα dΓ, Gs2(v) = Γg (gαsvα+ g3sv3) G2(θ, n)− g3sθα∂3vαG1(θ, n) dΓ, Gs3(v) =− Γg gs3θα∂3vαG3(θ, n) dΓ, (1.14)
where Γ = Γ0, Γg = Γ0g, and G1(θ, n), G2(θ, n), G3(θ, n) are bounded scalar functions of θ and n (the unit outer normal vector to the lateral boundary Γsfor s = 0). The space V is a subspace of [H02(]0, L[)]2×H1(Ω) =
[W02,2(]0, L[)]2× W1,2(Ω) defined by V = u∈ [H02(]0, L[)]2× H1(Ω) : v =v1(x3), v2(x3), v3(x1, x2, x3) , v3(x1, x2, x3) = v3(x3)− xα∂3vα(x3), with v3∈ H01(]0, L[) . (1.15)
We consider that V is equipped with the usual norm of [H1(Ω)]3. Finally, the closed convex K is defined by
K ={v ∈ V : v1(x3)≥ ψ(x3), in ]0, L[}. (1.16)
We remark that if we have considered the remodeling rate equation (1.8), then in (1.9) the ordinary differential equation would be the following
˙ ds= c(ds )e33(us) + b(ds) + 1 b3333(ds) e33(us)e33(us) +s −2 b3333(ds)θα∂33u s αe33(us)− c(ds)θα∂33usα +s2 1 b3333(ds) (θα∂33usα) (θβ∂33usβ), in Ω× (0, T ). (1.17)
In the sequel we represent by (u0, d0) the solution of problem (1.9) for s = 0, that is:
Find (u0, d0) such that: u0= (u0 1, u02, u03) : Ω× [0, T ] → R3 and d0: Ω× [0, T ] → R, u0(., t)∈ K ⊂ V, ad0(u0, v− u0)≥Ld0, v− u0, ∀v ∈ K, ˙ d0= b(d0) + c(d0)e 33(u0), in Ω× (0, T ), d0(x, 0) = d(x), in Ω, (1.18)
where ad0(., .) and Ld0(.) are independent of s and defined by ad0(z, v) = Ω 1 b3333(d0) e33(z) e33(v) dΩ, Ld0(v) = F0(v) + G0(v), (1.19) where F0(v) = Ω γ(ξ0+ Pη(d0)) (fαvα+ f3v3) dΩ, G0(v) = Γg gαvα+ g3v3 dΓ, (1.20)
for all z and v in V , with f = (fi) and g = (gi) independent of s. For the case where the remodeling rate
equation is defined by (1.17) then for s = 0 ˙ d0= b(d0) + c(d0)e 33(u0) + 1 b3333(d0) e33(u0)e33(u0), in Ω× (0, T ). (1.21) We also observe that because of the following Korn’s type inequality in the space V (cf. Ciarlet [1] or Valent [14]) ∃c > 0 : v2 [H1(Ω)]3 ≤ c e33(v)2L2(Ω), ∀v ∈ V, (1.22) where e33(v)2L2(Ω)=∂3v32L2(0,L)+ ω x2αdω ∂33vα2L2(0,L). (1.23) Thene33(.)L2(Ω)is a norm in the space V , equivalent to the usual norm induced in V by.[H1(Ω)]3. So in the sequel and for all v∈ V , we denote by vV the norme33(v)L2(Ω) or the normv[H1(Ω)]3. Moreover, V is a Hilbert space with the norme33(.)L2(Ω) and for each s, the bilinear form as0(., .) is continuous and elliptic in V (this statement is also a consequence of the condition imposed on the coefficient b3333(ds) in (1.7)), that is, there exist positive constants C1 and C2 independent of s, for all z and v in V and for all s∈ [0, δ], such that
as0(z, v)≤ C2e33(z)L2(Ω)e33(v)L2(Ω)= C2zVvV (continuity),
as0(v, v)≥ C1e33(v)L22(Ω)= C1v2V (ellipticity). (1.24)
The existence and uniqueness of solution to the family of bone remodeling rod models defined by (1.9) or (1.18) can be proved using the same arguments of Figueiredo and Trabucho [5] and also Monnier and Trabucho [9]. The proof of existence relies on Schauder’s fixed point theorem together with the Cauchy-Lipschitz-Picard theorem (used to solve the remodeling rate equation, for a fixed dispacement), the Stampacchia theorem (that is necessary to guarantee the existence of solution to the variational inequality, for a fixed change of volume fraction) and regularity results. The proof of uniqueness is based on arguments similar to those of Cowin and Nachlinger [3]. The next theorem summarizes this statement of existence and uniqueness.
Theorem 1.1 (Solution of (1.9)). Let s ∈ [0, δ] and we assume that, for each fixed ˆds, the unique solution ˆus
of the equilibrium problem
Find uˆs(., t)∈ K ⊂ V, such that:
adˆs(ˆus, v− ˆus)≥ Ldˆs, v− ˆus, ∀v ∈ K,
(1.25) has components with the regularity ˆus
α(., t)∈ W02,2(]0, L[)∩W3,2(]0, L[) and ˆu
s
3(., t)∈ W01,2(]0, L[)∩W2,2(]0, L[), for any t∈ [0, T ] (which implies that ˆus(., t)∈ W2,2(Ω)). Then, there exists a unique pair (us, ds) solution of
problem (1.9), verifying
us∈ C1([0, T ]; V ) and ds∈ C1[0, T ]; C0Ω . (1.26)
2. Partial Proof of Conditions (0.2) and (0.3)
We prove in this section that the conditions (0.2) and (0.3) are satisfied for sub-families {Asj}∞j=1 and
{Lsj}∞j=1, where sj ∈ [0, δ], of {As}s>0 and{Ls}s>0. Then, in Section 5 we conclude that (0.2) and (0.3) are
true for the entire families{As}s>0 and{Ls}s>0.
Theorem 2.1. Let (us, ds) and (u0, d0) be the solutions of problem (1.9) and (1.18), respectively. We assume that the conditions (1.7) are verified, and, for each s, ξs
0 = ξ0, fis = fi, gis = gi, where ξ0, fi and gi are
independent of s. Moreover we suppose that there exists a constant c > 0, such that us
C0([0,T ];W2,2(Ω)) ≤ c, for all s∈ [0, δ]. Then, for each t, there exists a subsequence of {(us, ds)(., t)}, denoted by {(usj, dsj)(., t)}, and
elements ¯u(., t)∈ V and ¯d(., t)∈ L2(Ω), such that, when sj → 0+ usj− u0 sj (., t) ¯u(., t) weakly in V, (2.1) e33 usj− u0 sj (., t) e33(¯u)(., t) weakly in L2(Ω), (2.2) dsj − d0 sj (., t) ¯d(., t) weakly in L2(Ω), (2.3) (usj− u0)(., t) −→ 0 strongly in V, (2.4) e33(usj − u0)(., t) −→ 0 strongly in L2(Ω), (2.5) (dsj − d0)(., t) −→ 0 strongly in L2(Ω), (2.6) e33(usj − u0) −→ 0 strongly in C0([0, T ]; C0( ¯Ω)), (2.7) dsj − d0 −→ 0 strongly in C0[0, T ]; C0( ¯Ω) . (2.8)
In addition the limit ¯d depends implicity on ¯u and is the solution of the following ordinary differential equation with respect to time
˙¯ d = c(d0) e 33(¯u) + ¯d c(d0)e 33(u0) + b(d0) − c(d0) θ α∂33u0α, ¯ d(x, 0) = 0, in Ω. (2.9)
Proof. The proof consists of four steps. The first two steps are preliminary results that prepare the proof of (2.1)–(2.8) in steps 3 and 4.
Step 1. There exist positive constants c1 and c2 independent of s, such that us
C0([0,T ];V )≤ usC0([0,T ];W2,2(Ω)) ≤ c1, ∀s ∈ [0, δ], (2.10) ds
C0([0,T ];L2(Ω))≤ c2, ∀s ∈ [0, δ]. (2.11)
The estimate (2.10) is a consequence of the hypotheses. Then, taking the integral, with respect to time in the remodeling rate equation we get
ds(x, t) = t 0 c(ds)e33(us) + b(ds)− s c(ds)θα∂33usα (x, r) dr + d(x). (2.12)
Then we immediately deduce (2.11) taking the L2(Ω) norm in the last equation and using (2.10) and (1.7). Step 2. There exist positive constants c3 and c4 independents of s, such that
us− us 0 C0([0,T ];V ) ≤ c3, ∀s ∈ [0, δ], (2.13) ds− d0 s C0([0,T ];L2(Ω)) ≤ c4, ∀s ∈ [0, δ]. (2.14)
Choosing v = u0 in problem (1.9) and v = us in problem (1.18) and subtracting the two corresponding
variational inequalities we obtain
ads(us, us− u0)− ad0(u0, us− u0)≤ Lds(us− u0)− Ld0(us− u0). (2.15) Dividing by s2and using the definitions of ads(., .) and Lds we have
as 0 us,u s− u0 s − as 0 u0,u s− u0 s s + as 0 u0,u s− u0 s − ad0 u0,u s− u0 s s ≤ (Fs 0+ Gs0)− (F0+ G0) s us− u0 s − as 1 us,u s− u0 s + (Fs 1 + Gs1) us− u0 s + o(s). (2.16)
Now using the estimates (2.10)–(2.11), the last inequality yields, for each t∈ [0, T ] as 0 us− u0 s (., t), us− u0 s (., t) ≤ cu s− u0 s (., t) V + c d s− d0 s (., t) L2(Ω) us− us 0(., t) V + o(s), (2.17)
where c and c are positive constants independent of s and t, and|o(s)| → 0, as s → 0+. Consequently, because of the ellipticity of as 0(., .), cf. (1.24), we have us− u0 s (., t) V ≤ cds− d0 s (., t) L2(Ω)) + c, (2.18)
where c and c are other positive constants independent of s and t. But subtracting the two remodeling rate equations in problems (1.9) and (1.18), and taking the integral with respect to time we obtain
(ds− d0)(x, t) = t 0 c(ds)e33(us− u0) + [c(ds)− c(d0)]e33(u0) +[b(ds)− b(d0)]− s c(ds)θ α∂33usα (x, r) dr, (2.19)
and therefore, using (1.7), the mean value theorem for the terms c(ds)− c(d0) and b(ds)− b(d0), and dividing by s, we obtain ds− d0 s (., t) L2(Ω) ≤ t 0 c1 us− u0 s (., r) V + c2 ds− d0 s (., r) L2(Ω) + c3 dr, (2.20)
where c1, c2 and c3 are other positive constants independent of s and t. Using now (2.18) and the integral Gronwall’s inequality (cf. Evans [4], p. 625) we have (2.14). Then, the property (2.13) is a consequence of (2.14) and (2.18).
Step 3. Because of the norm estimates (2.13)–(2.14) we directly obtain the weak convergences (2.1)–(2.3). The strong convergences (2.4)–(2.6) are a consequence of these weak convergences.
The strong convergence (2.7) is a consequence of (2.5) and the fact that ∂3(u
sj
3 − u03) and ∂33(u
sj
α − u0α) are
bounded in the space C0([0, T ]; W1,2(]0, L[)) and W1,2(]0, L[) is compactly imbedded in C0([0, L]).
Taking into account the definition of dsj− d0 given by (2.19), the strong convergence (2.8) is a consequence
of (2.7) and the integral’s Gronwall inequality.
Step 4. To prove (2.9) we consider in (2.19) s = sj and we divide by sj. Then for each t, when sj → 0+
c(dsj)(., t) −→ c(d0)(., t) strongly in C0(Ω), e33 usj− u0 sj (., t) e33(¯u)(., t) weakly in L2(Ω), c(dsj)− c(d0) sj e33(u0)(., t) ¯d c(d0) e33(u0)(., t) weakly in L2(Ω), b(dsj)− b(d0) sj (., t) ¯d b(d0)(., t) weakly in L2(Ω), ∂33usj α(., t) −→ ∂33u0α(., t) strongly in L2(Ω). (2.21)
Hence we conclude that, for each t, and for all v∈ L2(Ω) lim sj→0+ Ω dsj − d0 s (., t) v dΩ = Ω t 0 c(d0)e33(¯u) + ¯d c(d0) e33(u0) + ¯d b(d0)− c(d0) θα∂33u0α (x, r) dr v dΩ. (2.22)
Theorem 2.2. With the hypotheses of the previous Theorem 2.1, there exist A= Ad¯and L = Ld¯depending explicitly on ¯d and verifying, respectively, the conditions (0.2) and (0.3) for s = sj, that is,
lim sj→0+ Asj − A0 sj − A (., t) L(V ;V) = 0, (2.23) lim sj→0+ Lsj − L0 sj − L (., t) V = 0. (2.24)
For any u and v in V , A(., t)∈ L(V, V ) is defined by Au, v = − Ω b3333(d0) 1 b2 3333(d0) ¯ d e33(u)e33(v)dΩ + Ω −θα b3333(d0)(e33(u)∂33vα+ e33(v)∂33uα)dΩ + Ω 1 b3333(d0) e33(u)e33(v)divθdΩ, (2.25)
where b3333 is the first derivative of the scalar function b3333. The element L(., t)∈ V is defined by L(v) = Ω γ ¯d Pη(d0) (fαvα+ f3v3) dΩ + Ω γ(ξ0+ Pη(d0)) (fαvα+ f3v3) div θ− f3θα∂3vα dΩ + Γg (gαvα+ g3v3) G1(θ, n)− g3θα∂3vα dΓ, (2.26)
for any v in V , where Pη is the first derivative of the scalar function Pη.
Proof. We consider in the sequel s = sj. Using the definitions of Asand A0we obtain Asu, v − A0u, v s = ads(u, v)− ad0(u, v) s = as 0(u, v)− ad0(u, v) s + a s 1(u, v) + o(s), (2.27)
where|o(s)| tends to zero when s → 0+. The calculus of the limit as
1(u, v), when s→ 0+ is immediate. To compute the limit
as0(u,v)−ad0(u,v)
s , when
s→ 0+, we remark that the space C∞
0 ([0, L]) is dense in H02(]0, L[) and H01(]0, L[), for the norms .H2(]0,L[) and.H1(]0,L[), respectively. So by density, we only prove (2.25) for u∈ V and v = (v1, v2, v3− xα∂3vα)∈ V ,
such that, vα∈ C0∞([0, L]) and v3∈ C0∞([0, L]). Thus, for each t∈ [0, T ], when s → 0+, we obtain as0(u, v)− ad0(u, v) s = Ω 1 b3333(ds)− 1 b3333(d0) e33(u)e33(v) dΩ = Ω b3333(d0)− b3333(ds) ds− d0 b3333(ds) b3333(d0) −1 ds− d0 s e33(u) e33(v) dΩ ↓ − Ω b3333(d0) b3333(d0)−2d e¯ 33(u) e33(v) dΩ, (2.28) because e33(u) e33(v)∈ L2(Ω), b3333(d0)− b3333(ds) ds− d0 (., t)−→ b 3333(d0)(., t), in C0(Ω), (2.29)
and ds−ds 0(., t) converges weakly to ¯d(., t) in L2(Ω). Therefore (2.25) is proved. Applying the definitions of Lds and Ld0 we get
Lds(v)− Ld0(v) s = Fs 0(v) + Gs0(v)− F0(v)− G0(v) s + F s 1(v) + G s 1(v) + o(s), (2.30)
where|o(s)| tends to zero when s → 0+. So we obtain (2.26) by taking the limit in the latter expressions when s→ 0+, using the definitions of F0s, Gs0, F1s, Gs1, and remarking that
Fs 0 + Gs0− F0− G0 s (v)−→ Ω γ ¯dPη(d0)(fαvα+ f3v3)dΩ. (2.31) So we conclude that the conditions (0.2) and (0.3) are proved for s = sj.
3. Proof of Condition (0.5)
We show that condition (0.5) is verified, using a technique described in Sokolowski and Zolesio [12] p. 209, that consists in proving the polyhedricity of a modified constraint displacement set, and assuming an appropriate additional condition imposed to a non-negative Radon measure, as indicated in Proposition 3.4.
We consider the closed and convex subset S of H02(]0, L[) defined by
S ={ϕ ∈ H02(]0, L[) : ϕ(x3)≥ ψ(x3) in [0, L]} (3.1) and the operator
R : V −→ H02(]0, L[)
v = (v1, v2, v3) −→ R(v) = v1.
(3.2) It is clear that the constraint set K verifies
Moreover since R maps V onto H2
0(]0, L[) and 0∈ S ⊂ H02(]0, L[), we have KerR = KerR∩ K, where KerR = {v ∈ V : Rv = 0}. In addition V = KerR⊕(KerR)⊥, where (KerR)⊥={v ∈ V : a
d0(v, u) = 0, ∀u ∈ KerR}.
The next proposition defines the operator R−1 ∈ L(H2
0(]0, L[), (KerR)⊥), which is the right inverse of R, that is, R◦ R−1= idH02(]0,L[).
Proposition 3.1. The operator R−1 is defined by
R−1(ϕ) = (ϕ, v2, v3− x1∂3ϕ− x2∂3v2) = v + u, ∀ϕ ∈ H02(]0, L[), (3.4) where v = (0, v2, v3− x2∂3v2) is the element of KerR solution of the equation
ad0(v, z) =−ad0(u, z), ∀z ∈ KerR, (3.5)
and u = (ϕ, 0,−x1∂3ϕ).
Proof. We define R−1(ϕ) by (3.4), because R◦ R−1(ϕ) = ϕ and R−1(ϕ) must be in V . Moreover, as R−1(ϕ) must be in (KerR)⊥ we impose
ad0(R−1(ϕ), z) = 0, ∀z ∈ KerR. (3.6)
This is equivalent to find a v = (0, v2, v3− x2∂3v2) ∈ KerR, such that ad0(v + u, z) = 0, for all z ∈ KerR,
where u = (ϕ, 0,−x1∂3ϕ). Hence (3.5) is an immediate consequence of the linearity of ad0(., .) with respect to
the first component.
Obviously we can define a scalar product ((., .)) in H02(]0, L[) in the following way
((ζ, ξ)) = ad0(R−1ζ, R−1ξ), ∀ζ, ξ ∈ H02(]0, L[), (3.7)
and the orthogonal projection PS associated to this new scalar product is defined by
PS : H02(]0, L[)→ S ⊂ H02(]0, L[)
ξ→ PS(ξ)
(3.8) where ϕ = PS(ξ) is the unique solution of the following variational inequality
ϕ = PS(ξ)∈ S:
((ϕ− ξ, ζ − ϕ)) ≥ 0, ∀ζ ∈ S. (3.9)
Then, accordingly to Sokolowski and Zolesio [12], p. 209, for each t∈ [0, T ], the unique solution Π(Ld0)(., t) =
u0(., t) of the variational inequality u0(., t)∈ K ⊂ V, ad0(u0, v− u0)≥ Ld0, v− u0, ∀v ∈ K, (3.10) of problem (1.18), satisfies Π(Ld0)(., t) = Υ(Ld0)(., t) + R−1PS(Φ(Ld0))(., t),
with Υ(Ld0)(., t)∈ KerR, and R−1PS(Φ(Ld0))(., t)∈ (KerR)⊥.
(3.11)
For any l∈ V, the operator Υ : V→ KerR is defined by
Υ(l)∈ KerR ⊂ V :
ad0(Υ(l), z) =l, z, ∀z ∈ KerR ⊂ V,
and the operator Φ : V → H2 0(]0, L[) is defined as follows Φ(l)∈ H02(]0, L[) : ((Φ(l), ϕ)) =l, R−1ϕ, ∀ϕ ∈ H2 0(]0, L[). (3.13)
Due to the decomposition (3.11) and also because the mappings Υ, R−1 and Φ are linear and continuous we immediately conclude that, for each t∈ [0, T ], Π is conically differentiable at Ld0(., t), cf. (0.5), if and only if, PS is conically differentiable at Φ(Ld0)(., t).
We prove now that the orthogonal projection PS, with respect to the scalar product ((., .)) defined in (3.7),
is conical differentiable.
It is well known that the polyhedricity of the set S at a given point ϕ∈ S implies the conical differentiability of PS at ϕ. For convenience of the reader we include in the paper the next statement, that recalls the definition
of polyhedric set and the relation between polyhedricity and conical differentiability, applied to the set S and the projection PS (cf. Haraux [6], or Mignot [8], or Rao and Sokolowski [11]).
Proposition 3.2. The set S ⊂ H2
0(]0, L[) is polyhedric at ϕ∈ S, if for any ξ ∈ H02(]0, L[), such that ϕ = PS(ξ)
it follows
TS(ϕ)∩ [ϕ − ξ]⊥ = CS(ϕ)∩ [ϕ − ξ]⊥, (3.14)
where ⊥ denotes the orthogonal with respect to the inner product ((., .)), the closure is in the space H2 0(]0, L[), CS(ϕ) is the convex cone defined by
CS(ϕ) ={ζ ∈ H02(]0, L[) : ∃r>0, ϕ(x3) + rζ(x3)≥ ψ(x3) in ]0, L[}, (3.15) and TS(ϕ) = CS(ϕ) is the tangent cone to S at ϕ∈ S, that is, the closure in the space H02(]0, L[) of the convex
cone CS(ϕ).
If condition (3.14) is satisfied, for a pair (ϕ, ξ) in the space H2
0(]0, L[)× H02(]0, L[), with ϕ = PS(ξ), then for
all ζ∈ H2
0(]0, L[) and for s > 0 small enough
PS(ξ + sζ) = PS(ξ) + sPM(ζ) + o(s) and M = TS(ϕ)∩ [ϕ − ξ]⊥, (3.16)
where PM is the orthogonal projection on M , ando(s)H2(]0,L[)/s→ 0 as s → 0. The condition (3.16) means that PS is conical differentiable at ϕ∈ S.
Thus to conclude that PSis conical differentiable at a point ϕ∈ S it is enough to provide sufficient conditions
under which the set S is polyhedric at a point ϕ∈ S. These sufficient conditions are summarized in the next proposition.
Proposition 3.3. The set S is polyhedric at a point ϕ ∈ S, if the Radon measure µ defined by
((ϕ− PS(ϕ), ζ)) =−
L
0
ζdµ, ∀ζ ∈ C0∞(]0, L[) (3.17)
is non-negative and its support denoted by suppµ, that is a compact subset of [0, L] and verifies suppµ⊂ Ξψ=
{x3∈]0, L[: ϕ(x3) = ψ(x3)}, is admissible in the following sense
∀ζ ∈ H2
0(]0, L[), such that ζ = 0 C2− q.e on suppµ, implies that ζ∈ H2
0
]0, L[\suppµ . (3.18)
In consequence the set M defined in (3.16) is the following convex cone M =
ζ∈ H02(]0, L[\suppµ) : ζ(x3)≥ 0, C2− q.e. on Ξψ
(Note – we recall that a statement holds C2− q.e. if it holds except for a set of C2-capacity zero, where the C2-capacity of a compact set N , C2(N ), is defined by C2(N ) = inf
L
0 |∂33ζ(x3)|2dx3 : ζ ≥ 1 on N, 0 ≤ ζ∈ C0∞(]0, L[) .)
Proof. We first prove the two following statements i) and ii):
i) the scalar product ((., .)) is equivalent to the usual scalar product (., .) defined in H2
0(]0, L[) by
(ϕ, ξ) = L
0
∂33ϕ(x3) ∂33ξ(x3) dx3, ∀ϕ, ξ ∈ H02(]0, L[); (3.20) ii) the Radon measure µ defined in (3.17) is non-negative.
To prove i) we show that the norms.ad0 and.H02(]0,L[)associated to the scalar products ((., )) and (., .) defined by (3.7) and (3.20), respectively, are equivalent. For any ϕ∈ H2
0(]0, L[) we have (see Prop. 3.1) ϕ2 ad0 = ad0(R−1ϕ, R−1ϕ) = Ω 1 b3333(d0)(∂3v3− x1∂33ϕ− x2∂33v2) 2dΩ (3.21)
where v = (0, v2, v3− x2∂3v2)∈ KerR is such that
ad0(v, z) =−ad0(u, z), ∀z ∈ KerR, (3.22)
with u = (ϕ, 0,−x1∂3ϕ), and thus ϕ2
ad0 = ad0(u + v, u + v) = ad0(u, u) + 2ad0(u, v) + ad0(v, v). (3.23)
Choosing z = v in (3.22) and using condition (1.7) we obtain ϕ2
ad0=−ad0(v, v) + ad0(u, u)≤ ad0(u, u)
= Ω 1 b3333(d0) x21|∂33ϕ|2dΩ≤ cϕ2H2 0(]0,L[), (3.24)
where c is a positive constant. On the other hand, using again condition (1.7) and (1.24) we get ϕ2 ad0 ≥ C1e33(R−1ϕ)2L2(Ω) = C1∂3(v3− x1∂3ϕ− x2∂3v2)2L2(Ω) = C1 Ω (∂3v3)2dΩ + Ω x21(∂33ϕ)2dΩ + Ω x22(∂33v2)2dΩ ≥ C1 Ω x21(∂33ϕ)2dΩ = C ω x21dω ϕ2 H02(]0,L[), (3.25)
where C1 and C represent different positive constants. Thus the proof of i) is complete. To prove ii) it suffices to remark that for all ζ∈ C0∞(]0, L[) such that ζ≥ 0 in ]0, L[ we have
((ϕ− PS(ϕ), ζ)) = ((ϕ− PS(ϕ), ζ + PS(ϕ)− PS(ϕ)))
= ((ϕ− PS(ϕ), ξ− PS(ϕ)))≤ 0,
(3.26)
because of the definition of PS(ϕ) and the fact that ξ = ζ + PS(ϕ) belongs to S.
Due to the Properties i) and ii) and assuming that the set suppµ is admissible in the sense of (3.18), we finish the proof of this theorem, using exactly the same arguments as in Rao and Sokolowski [11].
The verification that the set suppµ is admissible in the sense of (3.18) is in general difficult (cf. also Pierre and Sokolowski [10] for the related subject of differentiability of projections and applications). Nevertheless there is a sufficient condition for which the suppµ is admissible, as described in the next Proposition 3.4 (cf. Rao and Sokolowski [11] for the proof of this statement). Therefore assembling this last comment with Proposition 3.3, we can state the following result concerning the conical differentiability of the projection PS at a point ϕ∈ S.
Proposition 3.4. If the support of the Radon measure µ defined in (3.17) is admissible in the sense of (3.18),
then, the set S is polyhedric at the point ϕ ∈ S, and, consequently, PS is conical differentiable at ϕ∈ S. In
particular, if the C1-capacity of the compact set suppµ is zero, that is C1(suppµ) = 0, where
C1(suppµ) = inf L 0 |∂3ζ(x3)|2dx3: ζ≥ 1 on suppµ, 0 ≤ ζ ∈ C0∞(]0, L[) ! , (3.27)
then suppµ is admissible.
Finally, assuming the hypothesis of the previous proposition 3.4, and using the decomposition (3.11) and (3.16) we conclude that, for each t∈ [0, T ], the operator Q(., t) in (0.5), which is the conical derivative of Π at Ld0(., t),
is defined by
Q(l)(., t) = Υ(l)(., t) + R−1PM (.,t)(Φ(l))(., t), ∀l ∈ V, (3.28) where for each t, the convex cone M (., t) depends on Ld0(., t) and the obstacle ψ, and is defined in (3.16) with
ϕ = Φ(Ld0)(., t), that is,
M (., t) = TS(Φ(Ld0)(., t))∩ [Φ(Ld0)(., t)− ξ]⊥, (3.29)
where Φ(Ld0)(., t) = PS(ξ), for some ξ∈ H02(]0, L[).
4. Partial Proof of Conditions (0.6) and (0.7)
In this section we prove that conditions (0.6) and (0.7) are satisfied for a sub-family {(usj, dsj)}∞ j=1 of
{(us, ds)}
s>0. In Section 5 we show that these two conditions are still verified for the all family{(us, ds)}s>0.
By Theorem 2.1 we know that there exists a subsequence that we denote by (usjs−u0
j , dsj−d0
sj )(., t) that
converges weakly to (¯u, ¯d)(., t) in V×L2(Ω), when sj→ 0+. Consequently, by Theorems 2.1 and 2.2, there exist
A = Ad¯and L = Ld¯that depend explicitly on ¯d and implicitly on ¯u. Using the Theorem 4.14 of Sokolowski and Zolesio [12] p. 178, combined with the expression (3.28) for Q(., t) (or equivalently Th. 4.30 of Sokolowski and Zolesio [12] p. 210) we conclude that, for all sj
usj(., t) = u0(., t) + s ju(., t) + o(sj), with u = Q(Ld¯− Ad¯u0) = Υ(L ¯ d− Ad¯u0) + R−1PM(Φ(Ld¯− Ad¯u0)), (4.1)
where o(sj)/sjV tends to zero when sj → 0+, and ¯d is the solution of the following ordinary differential
equation (cf. Th. 2.1) ˙¯ d = c(d0) e33(¯u) + ¯d c(d0)e33(u0) + b(d0) − c(d0) θ α∂33u0α ¯ d(x, 0) = 0, in Ω. (4.2)
Moreover from (4.1) and (2.1) we also conclude that ¯u = u. From (4.2) and (2.3) we deduce that ¯d = d and dsj(., t) = d0(., t) + s
jd(., t) + o(sj), (4.3)
where
Ωo(sj)vdΩ
sj tends to zero when sj → 0
+, for all v∈ L2(Ω). So the conditions (0.6) and (0.7) are proved for the subfamily of parameters s = sj.
5. Proof of Theorem 0.1
In this section we prove Theorem 0.1 with the hypotheses of Theorem 2.1 and the sufficient conditions of Proposition 3.3.
Observing (4.1), (4.2) and (4.3), and taking into account the results of Section 3 and also the Theorem 4.14 of Sokolowski and Zolesio [12] p. 178 (or equivalently Th. 4.30 of Sokolowski and Zolesio [12] p. 210), we realize that to prove conditions (0.2)–(0.3) and (0.6)–(0.7), and consequently to prove Theorem 0.1, it only remains to assure that the weak limit (¯u, ¯d)(., t) is unique. That is, for all s > 0, the sequence (us−us 0,ds−ds 0)(., t) converges weakly to (¯u, ¯d)(., t)∈ V × L2(Ω). This happens if the system defined by the second equation in (4.1) and (4.2) has a unique solution. In fact this is true, as stated and proved in the next theorem.
Theorem 5.1. The system Find (u, d)(., t)∈ V × L2(Ω) : u = Υ(Ld− Adu0) + R−1P M(Φ(Ld− Adu0)), ˙ d = c(d0) e 33(u) + d c(d0)e 33(u0) + b(d0) − c(d0) θα∂ 33u0α, d(x, 0) = 0, in Ω, (5.1)
has a unique solution (u, d)∈ C1([0, T ]; V )× C1([0, T ]; C0(Ω)).
Proof. The proof of existence is analogous to the proof of Theorem 1.1. It relies on Schauder’s fixed point theorem together with the Cauchy-Lipschitz-Picard theorem (used to solve the ordinary differential equation for a fixed u) and regularity results, concerning the first equation of (5.1). To prove that the solution of (5.1) is unique let (u, d) and (v, e) be two different solutions of (5.1). Then we have
u− v = Υ Ld− Le− (Ad− Ae)u0 +R−1 PM Φ(Ld− Adu0) + PM Φ(Le− Aeu0) . (5.2)
Taking the norm in V and using the continuity of the operators Υ, R−1, PM, Φ and the linearity of Υ, R−1, Φ,
we obtain for each t∈ [0, T ]
(u − v)(., t)V ≤ C(d − e)(., t)L2(Ω), (5.3)
where C is a positive constant. On the other hand, subtracting the two ordinary differential equations, and integrating in time (d− e)(x, t) = t 0 " c(d0) e33(u− v) + (d − e)c(d0)e33(u0) + b(d0)#(x, r)dr. (5.4)
Taking the L2(Ω), for each t∈ [0, T ], and using (5.3) we get (d − e)(., t)L2(Ω)≤ C
t
0
(d − e)(., t)L2(Ω)(x, r)dr, (5.5)
where C is positive constant independent of t. Applying now to (5.5) the integral Gronwall’s inequality we have
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